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3. (a) (i) A, B, C and DD lie on the circumference of the circle. Find the val...
Jul 21, 2024
Solution by Steps
step 1
Identify the given angles in the circle. Angle ∠ADB \angle ADB is given as 43 degrees and ∠BCD \angle BCD is labeled as t t
step 2
Use the property of cyclic quadrilaterals: Opposite angles of a cyclic quadrilateral sum to 180 degrees. Therefore, ∠ADB+∠BCD=180∘ \angle ADB + \angle BCD = 180^\circ
step 3
Substitute the given angle into the equation: 43∘+t=180∘ 43^\circ + t = 180^\circ
step 4
Solve for t t : t=180∘−43∘=137∘ t = 180^\circ - 43^\circ = 137^\circ
Answer
t=137∘ t = 137^\circ
Key Concept
Cyclic Quadrilateral Angle Sum
Explanation
In a cyclic quadrilateral, the sum of the opposite angles is always 180 degrees.
Problem 2: Finding the value of w w
step 1
Identify the given angles in the circle. Angle ∠XOY \angle XOY is labeled as w w and ∠YOZ \angle YOZ is given as 28 degrees
step 2
Use the property of angles subtended by the same arc: The angle at the center is twice the angle at the circumference. Therefore, ∠XOY=2×∠XYZ \angle XOY = 2 \times \angle XYZ
step 3
Since ∠YOZ \angle YOZ is given as 28 degrees, the angle at the circumference ∠XYZ \angle XYZ is half of this: ∠XYZ=28∘2=14∘ \angle XYZ = \frac{28^\circ}{2} = 14^\circ
step 4
Therefore, the angle at the center ∠XOY=2×14∘=28∘ \angle XOY = 2 \times 14^\circ = 28^\circ
Answer
w=28∘ w = 28^\circ
Key Concept
Angle at the Center and Circumference
Explanation
The angle subtended at the center of a circle is twice the angle subtended at the circumference by the same arc.
What is the relationship between the angles in a circle and the corresponding arcs
Solution by Steps
step 1
Identify the given angle in the circle. Angle ∠BAD \angle BAD is given as 37∘ 37^\circ
step 2
Use the property of angles subtended by the same arc in a circle. Angle ∠BED \angle BED is subtended by the same arc BD BD as ∠BAD \angle BAD . Therefore, ∠BED=∠BAD=37∘ \angle BED = \angle BAD = 37^\circ
step 3
For angle ∠BOD \angle BOD , use the property that the angle at the center is twice the angle at the circumference subtended by the same arc. Therefore, ∠BOD=2×∠BAD=2×37∘=74∘ \angle BOD = 2 \times \angle BAD = 2 \times 37^\circ = 74^\circ
step 4
For angle ∠BCD \angle BCD , use the property of cyclic quadrilaterals: Opposite angles of a cyclic quadrilateral sum to 180∘ 180^\circ . Therefore, ∠BCD=180∘−∠BAD=180∘−37∘=143∘ \angle BCD = 180^\circ - \angle BAD = 180^\circ - 37^\circ = 143^\circ
Answer
(a) ∠BED=37∘ \angle BED = 37^\circ because it is subtended by the same arc BD BD as ∠BAD \angle BAD .
(b) ∠BOD=74∘ \angle BOD = 74^\circ because the angle at the center is twice the angle at the circumference subtended by the same arc.
(c) ∠BCD=143∘ \angle BCD = 143^\circ because opposite angles of a cyclic quadrilateral sum to 180∘ 180^\circ .
Key Concept
Angles subtended by the same arc in a circle
Explanation
Angles subtended by the same arc at the circumference of a circle are equal, and the angle at the center is twice the angle at the circumference subtended by the same arc.
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